Ebook: Number Theory: Tradition and Modernization

Number Theory: Tradition and Modernization is a collection of survey and research papers on various topics in number theory. Though the topics and descriptive details appear varied, they are unified by two underlying principles: first, making everything readable as a book, and second, making a smooth transition from traditional approaches to modern ones by providing a rich array of examples.

The chapters are presented in quite different in depth and cover a variety of descriptive details, but the underlying editorial principle enables the reader to have a unified glimpse of the developments of number theory. Thus, on the one hand, the traditional approach is presented in great detail, and on the other, the modernization of the methods in number theory is elaborated. The book emphasizes a few common features such as functional equations for various zeta-functions, modular forms, congruence conditions, exponential sums, and algorithmic aspects.

Audience

This book is intended for researchers and graduate students in analytic number theory.

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This volume is a result of the author's four decades of research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the "Mathematics of Harmony," a new interdisciplinary direction of modern science. This direction has its origins in "The Elements" of Euclid and has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the "golden" algebraic equations, the generalized Binet formulas, Fibonacci and "golden" matrices), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational radices, Fibonacci computers, ternary mirror-symmetrical arithmetic, a new theory of coding and cryptography based on the Fibonacci and "golden" matrices).The book is intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science Includes an array of examples, this is a collection of survey and research papers on various topics in number theory. Presenting both the traditional and modern approaches, it emphasizes a few common features such as functional equations for various zeta-functions, modular forms, congruence conditions, exponential sums, and algorithmic aspects. Positive Finiteness of Number Systems; On a Distribution Property of the Residual Order of a (mod p)-- IV; Diagonalizing "Bad" Hecke Operators on Spaces of Cusp Forms; On the Hilbert-Kamke and the Vinogradov Problems in Additive Number Theory; The Goldbach-Vinogradov Theorem in Arithmetic Progressions; Densities of Sets of Primes Related to Decimal Expansion of Rational Numbers; Spherical Functions on p -Adic Homogeneous Spaces; On Modular forms of Weight (6n + 1)/5 Satisfying a Certain Differential Equation; Some Aspects of the Modular Relation